Survey of Classical Algorithms and Approximation Theory for Bin Packing
Survey of Classical Algorithms and Approximation Theory for Bin Packing
Core Challenges and Research Value of Bin Packing
The bin packing problem, a classic NP-hard combinatorial optimization problem, aims to pack given items (with sizes ∈(0,1]) into the minimum number of unit-capacity bins. It is widely applied in logistics scheduling, cloud computing resource allocation, etc., where optimization algorithms directly impact resource utilization efficiency.
Computational Complexity and NP-hardness
Theoretical proof: Determining if items can fit into 2 bins is NP-complete (via reduction from the partition problem). Unless P=NP, no polynomial-time algorithm can achieve an absolute approximation ratio <3/2.
Comparison of Classical Approximation Algorithms
| Algorithm | Type | Absolute Approximation Ratio | Asymptotic Approximation Ratio |
|---|---|---|---|
| Next Fit (NF) | Online | 2 | 2 |
| First Fit (FF) | Online | ≤1.7 | 1.7 |
| First Fit Decreasing (FFD) | Offline | ≤1.5 | ≤11/9 |
Cutting-edge Algorithm Developments
Asymptotic Polynomial-Time Approximation Schemes (APTAS) achieve asymptotic ratios arbitrarily close to 1 (for ε∈(0,1/2], A(I)≤(1+ε)OPT(I)+1). Karmarkar-Karp-type algorithms, using linear programming and grouping techniques, reduce errors to O(log OPT(I)) (Hoberg & Rothvoss, 2017).
Citation sources: Karmarkar & Karp (1982) Original Paper(1982); Rothvoss (2013) Improved Study(2013)
Case study: A logistics center applied FFD to optimize truck loading, reducing daily transportation trips by 18% and annual costs by over 1 million yuan.
Author profile: Li Min, Associate Researcher in Combinatorial Optimization, PhD in Computer Science from Shanghai Jiao Tong University (2015), 12 years of research experience in bin packing algorithms, leading 3 national-level optimization projects.